Numerical Weather Prediction (NWP) and climate models parametrize the effects of boundary-layer turbulence as a diffusive process, dependent on a diffusion coefficient, which appears as nonlinear terms in the governing equations. In the advection dominated zone of the boundary layer and in the free atmosphere, the air flow supports different wave motions, with the fastest being the sound waves. Time integrations of these terms, in both zones, need to be implicit otherwise they impractically restrict the stable time step sizes. At the same time, implicit schemes may lose accuracy compared to explicit schemes in the same level, which is due to dispersion error associated with these schemes. Furthermore, the implicit schemes need iterative approaches like the Newton-Raphson method. Therefore, the combination of implicit and explicit methods, called IMEX or semi-implicit, has extensively attracted attention. In the combined method, the linear part of the equation as well as the fast wave terms are treated by the implicit part and the rest is calculated by the explicit scheme. Meanwhile, minimizing the dissipation and dispersion errors can enhance the performance of time integration schemes, since the stability and accuracy will be restricted by these inevitable errors. Hence, the target of this thesis is to increase the stability range, while obtaining accurate solutions by using IMEX and semi-implicit time integration methods. Therefore, a comprehensive effort has been made toward minimizing the numerical errors to develop new Runge-Kutta schemes, in IMEX and semi-implicit forms, to temporally integrate the governing equations in the atmospheric field so that the stability is extended and accuracy is improved, compared to the previous schemes. At the first step, the A-stability and the Strong Stability Preserving (SSP) optimized properties were compared as two essential properties of the time integration schemes. It was shown that both properties attempt to minimize the dissipation and dispersion errors, but in two different aspects. The SSP optimized property focuses on minimizing the errors to increase the accuracy limits, while the A-stability property tries to extend the range of stability. It was shown that the combination of both properties is essential in the field of interest. Moreover, the A-stability property was found as an essential property to accelerate the steady state solutions. Afterward, the dissipation and dispersion errors, generated by three-stage second order IMEX Runge-Kutta scheme were minimized, while the proposed scheme, so called IMEX-SSP2(2,3,2) enjoys the A-stability and SSP properties. A practical governing equation set in the atmospheric field, so called compressible Boussinesq equations set, was calculated using the new IMEX scheme and the results were compared to one well-known IMEX scheme in the literature, i.e. ARK2(2,3,2), which is an abbreviation of Additive Runge-Kutta. Note that, the ARK2(2,3,2) was compared to various types of IMEX Runge-Kutta schemes and it was found as the more efficient scheme in the atmospheric fields (Weller et al., 2013). It was shown that the IMEX-SSP2(2,3,2) could improve the accuracy and extend the range of stable time step sizes as well. Through the van der Pol test case, it was shown that the ARK2(2,3,2) with L-stability property may decline to the first order in the calculation of stiff limit, while IMEX-SSP2(2,3,2), with A-stability property, is able to retain the assigned second order of accuracy. Therefore, it was concluded that the L-stability property, due to restrictive conditions associated with, may weaken the time integration’s performance, compared to the A-stability property. The ability of the IMEX-SSP2(2,3,2) was proved in solving different case, which is the inviscid Burger equation in spherical coordinate system by using a realistic initial condition dataset. In the next step, it was attempted to maximize the non-negativity property associated with the numerical stability function of three-stage third order Diagonally Implicit Runge-Kutta (DIRK) schemes. It was shown that the non-negativity has direct relation with non-oscillatory behaviors. Two new DIRK schemes with A- and L-stability properties, respectively, were developed and compared to the SSP(3,3), which obtains the SSP optimized property in the same class of DIRK schemes. The SSP optimized property was found to be more beneficial for the inviscid (advection dominated) flows, since in the von Neumann stability analysis, the SSP optimized property provides more nonnegative region for the imaginary component of the stability function. However, in most practical cases, i.e. the viscous (advection diffusion) flows, the nonnegative property is needed for both real and imaginary components of the stability function. Therefore, the SSP optimized property, individually, is not helpful, unless mixed with the A-stability property. Meanwhile, the A- and L-stability properties were compared as well. The intention is to find how these properties influence the DIRK schemes’ performances. The A-stability property was found as preserving the SSP property more than the L-stability property. Moreover, the proposed A-stable scheme tolerates larger Courant Friedrichs Lewy (CFL) number, while preserving the accuracy and non-oscillatory computations. This fact was proved in calculating different test cases, including compressible Euler and nonlinear viscous Burger equations. Finally, the time integration of the boundary layer flows was investigated as well. The nonlinearity associated with the diffusion coefficient makes the implicit scheme impractical, while the explicit scheme inefficiently limits the stable time step sizes. By using the DIRK scheme, a new semi-implicit approach was proposed, in which the diffusion coefficient at each internal stage is calculated by a weight-averaged combination of the solutions at current internal stage and previous time step, in which the time integration can benefit from both explicit and implicit advantages. As shown, the accuracy was improved, which is due to engaging the explicit solutions and the stability was extended due to taking advantages of implicit scheme. It was found that the nominated semi-implicit method results in less dissipation error, more accurate solutions and less CPU time usage, compared to the implicit schemes, and it enjoys larger range of stable time steps than other semi-implicit approaches in the literature.

This paper investigates the Runge-Kutta method of numerically integrating ordinary differential equations. An existence theorem is given insuring a solution to the differential equation, then the theorem is modified to yield an analytic solution. The derivation of the method itself is followed by an analysis of the inherent error.

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Os simuladores de vôo têm sido uma importante ferramenta para treinamento de pilotos e análise de vôo sem ter que se desembolsar grandes quantias monetárias, economizando combustível e evitando acidentes. Conseqüentemente, a demanda por simuladores de vôo tem aumentado tanto na indústria quanto na pesquisa. Com o intuito de futuramente construir um simulador de vôo, foi desenvolvido um projeto para elaboração de um software capaz de simular uma aeronave em vôo, do ponto de vista de dinâmica de vôo. O software SIMAERO foi desenvolvido na linguagem de programação C++ e simula a dinâmica de vôo de uma aeronave. Esta simulação consiste em resolver as equações de movimento da aeronave, utilizando o modelo matemático de equações diferenciais ordinárias proposto por ETKIN & REID, et al (1996). O modelo matemático é solucionado através do método de integração numérica Runge-Kutta de 4ª ordem conforme apresentado em CONTE (1977). Como parâmetros de entrada são informadas as seguintes características da aeronave: dados geométricos, dados aerodinâmicos e derivadas de estabilidade. Os resultados das simulações são apresentados em gráficos cartesianos e gravados em arquivos. Os gráficos são úteis para que possa ser feita uma posterior análise do comportamento da aeronave. Os arquivos gravados com os resultados das simulações podem ser utilizados em alguma aplicação futura, como sinas de entrada para uma plataforma de simulação, por exemplo. Neste trabalho será descrito como o SIMAERO foi desenvolvido e ao final serão apresentados alguns resultados obtidos. / Flight simulators have been an important tool for pilots training and for flight analyses, without having to spend a high quantity of money, saving gas and prevent accidents. Because of this, the demand for flight simulators has increased both in industry and in research centers. With the objective of in future build a flight simulator, a project to develop a software that is able to simulate the dynamics of flight of a flying aircraft was developed. The SIMAERO software was developed using C++ and its principal functionality is to simulate the dynamics of flight of an aircraft. This simulation basically is the solution of the system of motion equations of the aircraft, using the mathematical model described by ETKIN & REID, et al (1996). The mathematical model is solved using the 4th order Runge-Kutta numeric integration method, as presented in CONTE (1977). For the simulation, the geometric data, the aerodynamic data, and the dimensional derivates are passed to the software as input arguments. The results of the simulations are displayed as cartesians graphics and recorded as data files. The graphics are useful for visual analyses of the aircraft behavior, and the file, with the results of the simulation, can be used as input data for ground based simulator, for example. In this work, the development of the software SIMAERO will be presented, and then some results of the simulation of one aircraft will be shown.

Στο πρώτο κεφάλαιο περιγράφονται οι άμεσες μέθοδοι Runge – Kutta και ο προτεινόμενος ελεγκτής που είναι τύπου PI. Όταν το μέγεθος βήματος περιορίζεται από την αριθμητική ευστάθεια, ένα δυναμικό μοντέλο πρέπει να χρησιμοποιηθεί. Ένα τέτοιο μοντέλο παρήχθη και επαληθεύτηκε αριθμητικά για άμεσες μεθόδους Runge – Kutta. Εδώ περιγράφεται αυτό το δυναμικό μοντέλο. Στο δεύτερο κεφάλαιο περιγράφονται το πρόβλημα της επιλογής μεγέθους βήματος στα έμμεσα σχήματα Runge – Kutta και αναλύεται από μια άποψη ελέγχου ανατροφοδότησης. Οι ιδιότητες του νέου μοντέλου και της βελτιωμένης απόδοσης του νέου ελέγχου σφάλματος περιγράφονται χρησιμοποιώντας και ανάλυση και αριθμητικά παραδείγματα. Στο τρίτο κεφάλαιο αναλύεται και υλοποιείται σε περιβάλλον Μathematica η μη γραμμική διαφορική εξίσωση van der Ρol για μια σειρά από διαφορετικές τιμές της παραμέτρου ε. Επιπλέον σ’ αυτό το κεφάλαιο μελετάται η συμπεριφορά του συστήματος με την μέθοδο Runge-Kutta και με βάση τον ολοκληρωμένο αλγόριθμο του P ελέγχου βήματος. Στο τέταρτο κεφάλαιο περιγράφονται οι βασικές μέθοδοι για την επίλυση μη δύσκαμπτων συστημάτων συνήθων διαφορικών εξισώσεων από χαμηλές σε μεσαίες ανοχές. Εδώ δείχνεται πώς κατασκευάζονται μερικά ζεύγη χαμηλής τάξης χρησιμοποιώντας εργαλεία από την υπολογιστική άλγεβρα. Εστιάζεται η προσοχή μας πάνω σε μεθόδους που εξοπλίζονται με ανίχνευση τοπικού σφάλματος (για προσαρμοστικότητα στο μέγεθος βήματος) και με τη δυνατότητα να ανιχνευθεί η δυσκαμψία. Στο πέμπτο κεφάλαιο υλοποιείται σε περιβάλλον Mathematica η σύγκριση δυο αλγορίθμων ελέγχου (P και PI) του βήματος στην μη γραμμική διαφορική εξίσωση van der Pol υλοποιημένη σε RKclassic και RKdopri μέθοδο Στο έκτο κεφάλαιο μελετάται ένα πραγματικό σύστημα της μορφής y'=Α*y με βάση την μεθοδολογία που το προσομοιώνει η μέθοδος Runge-Kutta σε λογισμικό περιβάλλον Mathematica. Στο τελευταίο (έβδομο) κεφάλαιο γίνεται η σύγκριση του πραγματικού συστήματος και του προσομοιωμένου PI έλεγχου βήματος για τη μέθοδο Runge-Kutta. / -

Os simuladores de vôo têm sido uma importante ferramenta para treinamento de pilotos e análise de vôo sem ter que se desembolsar grandes quantias monetárias, economizando combustível e evitando acidentes. Conseqüentemente, a demanda por simuladores de vôo tem aumentado tanto na indústria quanto na pesquisa. Com o intuito de futuramente construir um simulador de vôo, foi desenvolvido um projeto para elaboração de um software capaz de simular uma aeronave em vôo, do ponto de vista de dinâmica de vôo. O software SIMAERO foi desenvolvido na linguagem de programação C++ e simula a dinâmica de vôo de uma aeronave. Esta simulação consiste em resolver as equações de movimento da aeronave, utilizando o modelo matemático de equações diferenciais ordinárias proposto por ETKIN & REID, et al (1996). O modelo matemático é solucionado através do método de integração numérica Runge-Kutta de 4ª ordem conforme apresentado em CONTE (1977). Como parâmetros de entrada são informadas as seguintes características da aeronave: dados geométricos, dados aerodinâmicos e derivadas de estabilidade. Os resultados das simulações são apresentados em gráficos cartesianos e gravados em arquivos. Os gráficos são úteis para que possa ser feita uma posterior análise do comportamento da aeronave. Os arquivos gravados com os resultados das simulações podem ser utilizados em alguma aplicação futura, como sinas de entrada para uma plataforma de simulação, por exemplo. Neste trabalho será descrito como o SIMAERO foi desenvolvido e ao final serão apresentados alguns resultados obtidos. / Flight simulators have been an important tool for pilots training and for flight analyses, without having to spend a high quantity of money, saving gas and prevent accidents. Because of this, the demand for flight simulators has increased both in industry and in research centers. With the objective of in future build a flight simulator, a project to develop a software that is able to simulate the dynamics of flight of a flying aircraft was developed. The SIMAERO software was developed using C++ and its principal functionality is to simulate the dynamics of flight of an aircraft. This simulation basically is the solution of the system of motion equations of the aircraft, using the mathematical model described by ETKIN & REID, et al (1996). The mathematical model is solved using the 4th order Runge-Kutta numeric integration method, as presented in CONTE (1977). For the simulation, the geometric data, the aerodynamic data, and the dimensional derivates are passed to the software as input arguments. The results of the simulations are displayed as cartesians graphics and recorded as data files. The graphics are useful for visual analyses of the aircraft behavior, and the file, with the results of the simulation, can be used as input data for ground based simulator, for example. In this work, the development of the software SIMAERO will be presented, and then some results of the simulation of one aircraft will be shown.

We use the social force model to study the behaviour of two crowds of pedestrians in a bi-directional flow. There are two main goals of the project. The first goal is to study the effects of perception anisotropy on lane formation of the two crowds. The perception anisotropy models the fact that people actually do not perceive their surroundings equally well in all directions, i.e they have a field of vision. The second goal is to develop pedestrian viscosity indices to characterize the motion of the crowds. Our concepts of viscosity indices estimate how fluid the motion of the crowds are. We develop two viscosity indices. A lane viscosity index which gives information of the flow on large timescales, and a space-dependent viscosity index which can pinpoint where in space the motion is less fluid. Our results indicate that there is a small correlation between the perception anisotropy and the lane formation of the two crowds. The two viscosity indices work as intended but more refinement is needed to cope with simultaneous space-time changes.

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En esta tesis se estudia la dinámica del control biológico mediante un modelo matemático de cadena alimenticia simple de tres niveles tróficos. Este modelo matemático esta basado en un modelo depredador presa con respuesta funcional Holling tipo II razón dependiente, incluyendo un depredador superior para obtener un sistema de tres ecuaciones diferenciales ordinarias. Para el cual se estudia la existencia y unicidad, invarianza y acotación de las soluciones. La dinámica del control biológico es estudiada de forma local y asintótica, analizando las condiciones para la coexistencia de las tres especies así como también los escenarios de extinción total y parcial del sistema, de donde vemos cuando tiene o no tiene éxito el control biológico. Los resultados obtenidos fueron contrastados con sus respectivas simulaciones realizadas en un programa desarrollado en la tesis el cual aproxima las soluciones utilizando el método de Runge - Kutta de cuarto orden. PALABRAS CLAVE: DEPREDADOR - PRESA, CADENA ALIMENTICIA, CONTROL BIOLÓGICO, ESCENARIOS DE EXTINCIÓN, ESTABILIDAD ASINTÓTICA, RUNGE-KUTTA. / --- In this work, the dynamic of biological control is studied using a mathematical model of simple food chain of three trophic levels. This mathematical model is based on a predator prey model with holling type II functional response rate dependent. including a top predator described by a system of three ordinary differential equations. We study the existence and uniqueness, invariance and boundednees of solutions.